1 Selection rules

Selection rules that differentiate between formally allowed and forbidden transitions can be derived from the theoretical expression for the transition moment. A transition with a vanishing transition moment is referred to as being forbidden and should have zero intensity. But it should be remembered that the transition moment of an allowed transition, although nonvanishing, can still be very small, whereas a forbidden transition may be observed in the spectrum with finite intensity if the selection rule is relaxed by an appropriate perturbation. The most important example are vibration-ally induced transitions, which will be discussed later. (Cf. Section 1.3.4.) Other effects such as solvent perturbations may play a significant role also. Finally, since a series of approximations is necessary in order to derive the selection rules, they can be obeyed only within the limits of validity of these approximations. 

For instance, from the approximations introduced into the theoretical treatment of the radiation field it follows that only one-photon processes are allowed. However, very intense radiation fields, especially those produced by lasers, can cause simultaneous absorption of two photons, thus making it possible to reach molecular states that are not accessible from the ground state via one-photon absorption. Quite often, the only other evidence for the existence of these states is indirect, and two-photon absorption spectroscopy is complementary to conventional one-photon spectroscopy. (Cf. Section 1.3.6.) 

The spin selection rule is a consequence of the fact that the electric dipole and quadrupole moment operators do not operate on spin. Integration over the spin variables then always yields zero if the spin functions of the two states 0 and are different, and an electronic transition is spin allowed only if the multiplicities of the two states involved are identical. As a result, singlet-triplet absorptions are practically inobservable in the absorption spectra of hydrocarbons, or for that matter, other organic compounds without heavy atoms. Singlet-triplet excitations are readily observed in electron energy loss spectroscopy (EELS), which obeys different selection rules (Kuppermann et al., 1979). 

Strictly speaking, however, the spin angular momentum and its components are not constants of motion in nonlinear molecules, and the classification of states by multiplicity is therefore only approximate. Spin-orbit coupling is the most important of the terms in the Hamiltonian that cause a mixing of zero-order pure multiplicity states. The interaction between the spin angular momentum of an electron and the orbital angular momentum of the same electron causes the presence of a minor term in the Hamiltonian, which may be written as 

tj = x pj is the orbital angular momentum operator of the electron j, is the vector pointing from nucleus p to electron j, and the sums run over all electrons j and all nuclei p. Spin-orbit coupling is particularly significant in the presence of atoms of high atomic number ( heavy atom effect ). 

The selection rules are restrictions imposed on the quantum transitions, because of the laws of conservation of angular momentum and parity [59], In the case of IR spectroscopy, within the frame of the harmonic approximation, the applicable rules are the electric dipole selection rules. That is, when the expression in Equation 4.19 has a finite value, the transition is allowed, and when this expression is zero the transition is forbidden. In the Raman case, when one of the integrals given by Equation 4.23 is different from zero, the normal vibration associated is Raman-active. 

The Physical Chemistry of Materials Energy and Environmental Applications 

Since the wave functions included in Equations 4.19a and 4.23 are invariant under symmetry operations, to calculate the selection rules, it is necessary to take into account the symmetry of the absorption system [56], 

Symmetry operations are studied with the help of group theory, which discusses the set of operations that satisfies the following four conditions [60]  

Consequently, in the case of molecules, the symmetry operations that form the point groups convert the molecule into self-coincidence [10]. 

The selection rules [118] for infrared absorption and Raman scattering in crystals can be derived by taking into consideration conservation of energy, con- 

In a two-phonon process which involves an internal mode of vibration for which the frequency p varies little with wave vector, the combined density of states approximates, at least in a cubic crystal, the density of vibrational states spectrum [see Equation (6)]. This is important for a discussion of the absorption that occurs on the high- and low-frequency sides of many internal-mode absorption lines. 

From the above discussion when dpjdq, or more rigorously, VqPy is zero or has a slope discontinuity, there are likely to be slope discontinuities in the combined density of states, as revealed by infrared and Raman spectra of two-phonon processes. Points in the Brillouin zone where each of the components of VqP = 0 are known as critical points. The intensity of infrared absorption or Raman scattering depends upon quantum mechanical matrix elements which are, in general, not simple to evaluate. However, by using symmetry considerations and group theoretical methods, the various modes can be assigned as infrared or Raman active. 

in Developments in Inorganic Nitrogen Chemistry, C. B. Colburn, ed., Elsevier, Amsterdam, 1967, Chapter 11. 

Herzberg, Molecular Spectra and Molecular Structure. II. Infrared and Raman Spectra of Polyatomic Molecules, Van Nostrand, New York, 1945. 

The selection rules allow us to determine the number of bands to expect in vibrational spectra, since only vibrations belonging to certain irreducible representations lead to bands in the spectra. 

In the second part (Sections 6.4-6.7) we will consider what the vibrations belonging to a given irreducible representation look like . This involves the construction of linear combinations of the basis of atomic movements that are consistent with the characters of the irreducible representation. These combinations are known as symmetry adapted linear combinations (SALCs). SALCs are a general way to visualize the molecular properties that correspond to the objects with the irreducible symmetries identified by the reduction formula. The method we shall use to obtain these SALCs is the projection operator, which is introduced in Section 6.6 and will also be employed in Chapter 7 to find molecular orbitals. 

Finally, in Section 6.8, examples of vibrational analysis and its use in differentiating 

Selection rules are used in spectroscopy to determine whether a transition between two energy states within a molecule will show up in its spectrum. To be seen the transition has to be able to couple to the light which is used as a probe. This coupling is controlled by integrals over the initial and final states for the transition and the appropriate molecular property for the type of spectroscopy. 

IR absorption occurs when the transition between two vibrational states of a molecule has an energy matching the photon energy of the probe radiation and the transition causes a change in the molecular dipole moment. If there is no change in molecular dipole moment during the vibration, then there will be no absorption and we say that the mode is not allowed by the selection rules. 

Since these selection rules must be strictly obeyed, why do many cl-block metal complexes exhibit d-d bands in their electronic spectra  

As we saw in Section 21.6, electronic energy levels are labelled with term symbols. For the most part, we shall use the simplified form of these labels, omitting the J states. Thus, the term symbol is written in the general form  

Electronic transitions between energy levels obey the following selection rules. 

Transitions may occur from singlet to singlet, or from triplet to triplet states, and so on, but a change in spin multiplicity is forbidden. 

Laporte selection rule-. There must be a change in parity allowed transitions g u 

There are two major selection rules for absorption transitions  

Intersystem crossing (i.e. crossing from the first singlet excited state Si to the first triplet state Tj) is possible thanks to spin-orbit coupling. The efficiency of this coupling varies with the fourth power of the atomic number, which explains why intersystem crossing is favored by the presence of a heavy atom. Fluorescence quenching by internal heavy atom effect (see Chapter 3) or external heavy atom effect (see Chapter 4) can be explained in this way. 

Symmetry-forbidden transitions. A transition can be forbidden for symmetry reasons. Detailed considerations of symmetry using group theory, and its consequences on transition probabilities, are beyond the scope of this book. It is important to note that a symmetry-forbidden transition can nevertheless be observed because the molecular vibrations cause some departure from perfect symmetry (vibronic coupling). The molar absorption coefficients of these transitions are very small and the corresponding absorption bands exhibit well-defined vibronic bands. This is the case with most n — n transitions in solvents that cannot form hydrogen bonds (e 100-1000 L mol-1 cm-1). 

Values of range from close to zero (a very weak absorption) to 10000dm mon cm (an intense absorption). 

Some important points (for which explanations will be given later in the section) are that the electronic spectra of  

The relative intensities of absorption bands are governed by a series of selection rules. On the basis of the symmetry and spin multiplicity of ground and excited electronic states, two of these rules may be stated as follows  

Transitions between states of the same parity (symmetry with respect to a center 

Transitions between states of different spin multiplicities are forbidden. For example, transitions between 2 states are spin-allowed, but between 

These rules would seem to rule out most electronic transitions for transition metal complexes. However, many such complexes are vividly colored, a consequence of various mechanisms by which these rules can be relaxed. Some of the most important of these mechanisms are as follows  

Spin-orbit coupling in some cases provides a mechanism of relaxing the second selection rule, with the result that transitions may be observed from a ground state of one spin multiplicity to an excited state of different spin multiphcity. Such absorption bands for first-row transition metal complexes are usually very weak, with typical molar absorptivities less than 1 L moF cm For complexes of second-and third-row transition metals, spin-orbit coupling can be more important. 

Tetrahedral complexes often absorb more strongly than octahedral complexes of the 

operator is a one-electron operator which is even under time reversal, and non-totally symmetric in spin and orbit space. The trace of the spin-orbit coupling matrix for the t2g-shell thus vanishes. As a result the s.o.c. operator is found to transform as the MK = 0 component of a pure quasi-spin triplet (Cf. Eq. 26). Application of the selection rule in Eq. 28 shows that allowed matrix elements must involve a change of one unit in quasi-spin character, i.e. AQ = 1. Since 4S and 2D are both quasi-spin singlets while 2P is a quasispin triplet, s.o.c. interactions will be as follows  

In a d-only approximation a further selection rule can be invoked, based on pseudo-angular momenta. This requires a comparison between the actions of the pseudo and true angular momentum operators in the basis of the real 

Although the two tables are quite different, a remarkable coincidence is observed if one restricts attention to the matrix elements within the t2g-shell [3], Indeed for these matrix elements one has  

In the following section we will see how these rules materialize in the actual spectrum. 

They further defined two possible pathways for such reactions 

Marvell, E. N. Thermal Electrocyclic Reactions Academic Press New York, 1980. 

The new r bond formed in the cyclic structure is described in terms of a localized r orbital and a localized cr antibonding orbital formed by overlap of a pair of sp hybrid orbitals that result from rotation of the two terminal p orbitals in the 7i system. 

If the cyclobutene has an alkyl substituent at C3, both Z and E isomers are possible from the conrotatory pathway. Both steric effects and electronic effects of substituents elsewhere can influence the product distribution. See the discussion on page 757. 

Disrotatory (left) and conrotatory (right) electrocyclic conversion of 1,3-butadiene to cyclobutene. 

Suppose each state wavefunction in Eq. (4.3) can be written as a simple product of space-only and spin-only parts then Q is given by 

Regardless of the nature of the space parts, Q vanishes if V spin V spm- If Q vanishes, so does /. Thus we have the so-called spin-selection rule which denies the possibility of an electronic transition between states of different spin-multiplicity and we write AS = 0 for spin-allowed transitions. Expressed in different words, transitions between states of different spin are not allowed because light has no spin properties and cannot, therefore, change the spin. 

Now consider a transition between states of the same spin. The spin overlap integral , ij/ pin I V spin , in Eq. (4.5) is non-zero if all relevant functions are normalized, it is unity. So we turn our attention to the space part of Q (Eq. 4.6). 

Let us enquire about the electric dipole transition moment between two d orbitals as expressed in Eq. (4.7). 

The d orbitals are centrosymmetric and are of g symmetry. The light operator, being dipolar, is of u symmetry. The symmetry of the whole function under the integral sign in (4.7) - that is, for the product d r d- is g x m x g, namely u. The integral over all volumes of a m function vanishes identically. Since Q in (4.7) then vanishes, so does the intensity /. In short, d-d transitions are disallowed. 

As noted in Section 5.4, the transition probability between two electronic states is proportional to the square of the modulus of the matrix element 

Example 10.1-1 Discuss the transitions which give rise to the absorption spectrum of benzene. 

A preliminary analysis of the absorption spectrum was given in Example 5.4-1 as an illustration of the application of the direct product (DP) rule for evaluating matrix elements, but the analysis was incomplete because at that stage we were not in a position to deduce the symmetry of the electronic states from electron configurations, so these were merely stated. A more complete analysis may now be given. The molecular orbitals (MOs) 

The vibrational wave function, as any wave function, must form a basis for an irreducible representation of the molecular point group [3], 

The total vibrational wave function, Vl/v, can be written as the product of the wave functions where 4T is the wave function 

In general, at any time, each of the normal modes may be in any state. There is, however, a situation when all the normal modes are in their ground states and only one of them gets excited into the first excited state. Such a transition is called a fundamental transition. The intensity of the fundamental transitions is much higher than the intensity of the other kinds of transitions. Therefore, these are of particular interest. 

The vibrational wave function of the ground state belongs to the totally symmetric irreducible representation of the point group of the molecule. The wave function of the first excited state will belong to the irreducible representation to which the normal mode undergoing the particular transition belongs. 

A fundamental transition will occur only if one of the following integrals has nonzero value  

Optical absorption in a medium can take place because of the existence of electric or magnetic dipole moments associated with atomic, molecular or crystal entities. Unless otherwise specified, only electric dipoles are considered here. In quantum mechanics, the condition related to the dipole moment for discrete optical absorption appears in terms of transition probability between the initial and final states. It can be formally expressed as the modulus squared of a matrix element involving the wave functions and Ff of the initial and final states and the electric dipole operator, which reduces, within a proportionality factor to the general displacement coordinate ra  

The symmetry of an isolated atom is that of the full rotation group R+ (3), whose irreducible representations (IRs) are D where j is an integer or half an odd integer. An application of the fundamental matrix element theorem [22] tells that the matrix element (5.1) is non-zero only if the IR DW of Wi is included in the direct product x of the IRs of ra and f. The components of the electric dipole transform like the components of a polar vector, under the IR l)(V) of R+(3). Thus, when the initial and final atomic states are characterized by angular momenta Ji and J2, respectively, the electric dipole matrix element (5.1) is non-zero only if D(Jl) is contained in Dx D(j 2 ) = D(J2+1) + T)(J2) + )(J2-i) for j2 1 This condition is met for = J2 + 1, J2, or J2 — 1. However, it can be seen that a transition between two states with the same value of J is allowed only for J 0 as DW x D= D( D(°) is the unit IR of R+(3)). For a hydrogen-like centre, when an atomic state is defined by an orbital quantum number , this can be reduced to the Laporte selection rule A = 1. This is of course formal, as it will be shown that an impurity state is the weighted sum of different atomic-like states with different values of but with the same parity P = ( —1) These states are represented by an atomic spectroscopy notation, with lower case letters for the values of (0, 1, 2, 3, 4, 5, etc. correspond to s, p, d, f, g, h, etc.). The impurity states with P = 1 and -1 are called even- and odd-parity states, respectively. For the one-valley EM donor states, this quasi-atomic selection rule determines that the parity-allowed transitions from Is states are towards np (n 2), n/ (n 4), nh (n 6), or nj (n 8) states. For the acceptor states in cubic semiconductors, the even- and odd-parity states labelled by the double IRs T of Oh or Td are indexed by + or respectively, and the parity-allowed transition take place between Ti+ and 

Ej states.1 As already mentioned, these series of donor and acceptor transitions are somewhat analogous to the Lyman series for the H atom, and hence the corresponding impurity spectra are sometimes called Lyman spectra. 

The time-dependent Schrodinger equation for an atom in an electromagnetic field reads 

The vector potential of a plane wave is periodic and, if terms in A2 are neglected, the perturbation due to the field corresponds to a matrix element 

For small k r, which is satisfactory at long wavelengths, but breaks down towards the X-ray range, the exponential can be replaced by unity and higher order terms are neglected. In this case 

2 For a discussion of these, we refer the reader to standard texts, e.g. [149]. 

Since —ev r j is the component of the electric dipole moment connecting states i and j, this is called the electric dipole approximation. 

Spectroscopy is a powerful tool for studying matter. The treatment in this text cannot do the topic justice—series of books are written on just the topics in the next three chapters. However, the following material should give you some idea of how spectroscopy aids our understanding of atoms and molecules. 

Equation 14.1 is written in the original form of the Bohr frequency condition The difference in energy of the two quantum states equals the energy of the photon, which equals hv. 

Remember, however, that wavefunctions have symmetry, and so do operators. The light that causes the system to go from one state to another (either by absorption or emission) can be assigned an irreducible representation from the point group of the system of interest. Quantum mechanics defines a specific expression, called a transition moment, to which the irreducible representations can be applied. For an absorption or emission of a photon, the transition moment M is defined as 

We can use the conclusions of the previous chapter on symmetry at this point. In order for the integral in equation 14.2 to have a nonzero numerical value, the irreducible representations of the three components of the integrand must contain the totally symmetric irreducible representation of the point group of the system, usually labeled Aj. That is. 

Unless otherwise noted, all art on this page is Cengage Learning 2014 

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This spectrum is called a Raman spectrum and corresponds to the vibrational or rotational changes in the molecule. The selection rules for Raman activity are different from those for i.r. activity and the two types of spectroscopy are complementary in the study of molecular structure. Modern Raman spectrometers use lasers for excitation. In the resonance Raman effect excitation at a frequency corresponding to electronic absorption causes great enhancement of the Raman spectrum. 

The solutions can be labelled by their values of F and m.p. We say that F and m.p are good quantum. num.bers. With tiiis labelling, it is easier to keep track of the solutions and we can use the good quantum numbers to express selection rules for molecular interactions and transitions. In field-free space only states having the same values of F and m.p can interact, and an electric dipole transition between states with F = F and F" will take place if and only if... 

The vanishing integral rule is not only usefi.il in detemiining the nonvanishing elements of the Hamiltonian matrix H. Another important application is the derivation o selection rules for transitions between molecular states. For example, the hrtensity of an electric dipole transition from a state with wavefimction "f o a... 

Quack M 1977 Detailed symmetry selection rules for reactive collisions Mol. Phys. 34 477-504... 

Cordonnier M, Uy D, Dickson R M, Kew K E, Zhang Y and Oka T 2000 Selection rules for nuclear spin modifications in ion-neutral reactions involving Hg" J. Chem. Phys. 113 3181-93... 

Electronic spectra are almost always treated within the framework of the Bom-Oppenlieimer approxunation [8] which states that the total wavefiinction of a molecule can be expressed as a product of electronic, vibrational, and rotational wavefiinctions (plus, of course, the translation of the centre of mass which can always be treated separately from the internal coordinates). The physical reason for the separation is that the nuclei are much heavier than the electrons and move much more slowly, so the electron cloud nonnally follows the instantaneous position of the nuclei quite well. The integral of equation (BE 1.1) is over all internal coordinates, both electronic and nuclear. Integration over the rotational wavefiinctions gives rotational selection rules which detemiine the fine structure and band shapes of electronic transitions in gaseous molecules. Rotational selection rules will be discussed below. For molecules in condensed phases the rotational motion is suppressed and replaced by oscillatory and diflfiisional motions. 

Transition intensities are detennined by the wavefiinctions of the initial and final states as described in the last sections. In many systems there are some pairs of states for which tire transition moment integral vanishes while for other pairs it does not vanish. The temi selection rule refers to a simnnary of the conditions for non-vanishing transition moment integrals—hence observable transitions—or vanishing integrals so no observable transitions. We discuss some of these rules briefly in this section. Again, we concentrate on electric dipole transitions. 

Atoms have complete spherical synnnetry, and the angidar momentum states can be considered as different synnnetry classes of that spherical symmetry. The nuclear framework of a molecule has a much lower synnnetry. Synnnetry operations for the molecule are transfonnations such as rotations about an axis, reflection in a plane, or inversion tlnough a point at the centre of the molecule, which leave the molecule in an equivalent configuration. Every molecule has one such operation, the identity operation, which just leaves the molecule alone. Many molecules have one or more additional operations. The set of operations for a molecule fonn a mathematical group, and the methods of group theory provide a way to classify electronic and vibrational states according to whatever symmetry does exist. That classification leads to selection rules for transitions between those states. A complete discussion of the methods is beyond the scope of this chapter, but we will consider a few illustrative examples. Additional details will also be found in section A 1.4 on molecular symmetry. 

We now turn to electronic selection rules for syimnetrical nonlinear molecules. The procedure here is to examme the structure of a molecule to detennine what synnnetry operations exist which will leave the molecular framework in an equivalent configuration. Then one looks at the various possible point groups to see what group would consist of those particular operations. The character table for that group will then pennit one to classify electronic states by symmetry and to work out the selection rules. Character tables for all relevant groups can be found in many books on spectroscopy or group theory. Ftere we will only pick one very sunple point group called 2 and look at some simple examples to illustrate the method. 

The applications to selection rules work as follows. Intensities depend on the values of the transition moment integral of equation (Bl.l.lT... 

Analogous considerations can be used for magnetic dipole and electric qiiadnipole selection rules. The magnetic dipole operator is a vector with tln-ee components that transfonn like R, R and R. The electric... 

Most stable polyatomic molecules whose absorption intensities are easily studied have filled-shell, totally synuuetric, singlet ground states. For absorption spectra starting from the ground state the electronic selection rules become simple transitions are allowed to excited singlet states having synuuetries the same as one of the coordinate axes, v, y or z. Other transitions should be relatively weak. 

Often it is possible to resolve vibrational structure of electronic transitions. In this section we will briefly review the symmetry selection rules and other factors controlling the intensity of individual vibronic bands. 

The selection rule for vibronic states is then straightforward. It is obtained by exactly the same procedure as described above for the electronic selection rules. In particular, the lowest vibrational level of the ground electronic state of most stable polyatomic molecules will be totally synnnetric. Transitions originating in that vibronic level must go to an excited state vibronic level whose synnnetry is the same as one of the coordinates, v, y, or z. 

One of the consequences of this selection rule concerns forbidden electronic transitions. They caimot occur unless accompanied by a change in vibrational quantum number for some antisynnnetric vibration. Forbidden electronic transitions are not observed in diatomic molecules (unless by magnetic dipole or other interactions) because their only vibration is totally synnnetric they have no antisymmetric vibrations to make the transitions allowed. 

The synnnetry selection rules discussed above tell us whether a particular vibronic transition is allowed or forbidden, but they give no mfonnation about the intensity of allowed bands. That is detennined by equation (Bl.1.9) for absorption or (Bl.1.13) for emission. That usually means by the Franck-Condon principle if only the zero-order tenn in equation (B 1.1.7) is needed. So we take note of some general principles for Franck-Condon factors (FCFs). 

A very weak peak at 348 mn is the 4 origin. Since the upper state here has two quanta of v, its vibrational syimnetry is A and the vibronic syimnetry is so it is forbidden by electric dipole selection rules. It is actually observed here due to a magnetic dipole transition [21]. By magnetic dipole selection rules the A2- A, electronic transition is allowed for light with its magnetic field polarized in the z direction. It is seen here as having about 1 % of the intensity of the syimnetry-forbidden electric dipole transition made allowed by... 

If the experunental technique has sufficient resolution, and if the molecule is fairly light, the vibronic bands discussed above will be found to have a fine structure due to transitions among rotational levels in the two states. Even when the individual rotational lines caimot be resolved, the overall shape of the vibronic band will be related to the rotational structure and its analysis may help in identifying the vibronic symmetry. The analysis of the band appearance depends on calculation of the rotational energy levels and on the selection rules and relative intensity of different rotational transitions. These both come from the fonn of the rotational wavefunctions and are treated by angnlar momentum theory. It is not possible to do more than mention a simple example here. 

They are caused by interactions between states, usually between two different electronic states. One hard and fast selection rule for perturbations is that, because angidar momentum must be conserved, the two interacting states must have the same /. The interaction between two states may be treated by second-order perturbation theory which says that the displacement of a state is given by... 

Spectroscopists observed that molecules dissolved in rigid matrices gave both short-lived and long-lived emissions which were called fluorescence and phosphorescence, respectively. In 1944, Lewis and Kasha [25] proposed that molecular phosphorescence came from a triplet state and was long-lived because of the well known spin selection rule AS = 0, i.e. interactions with a light wave or with the surroundings do not readily change the spin of the electrons. 

Before presenting the quantum mechanical description of a hannonic oscillator and selection rules, it is worthwhile presenting the energy level expressions that the reader is probably already familiar with. A vibrational mode v, witii an equilibrium frequency of (in wavenumbers) has energy levels (also in... 

The electric dipole selection rule for a hannonic oscillator is Av = 1. Because real molecules are not hannonic, transitions with Av > 1 are weakly allowed, with Av = 2 being more allowed than Av = 3 and so on. There are other selection niles for quadnipole and magnetic dipole transitions, but those transitions are six to eight orders of magnitude weaker than electric dipole transitions, and we will therefore not concern ourselves with them. 

CAHRS and CSHRS) [145, 146 and 147]. These 6WM spectroscopies depend on (Im for HRS) and obey the tlnee-photon selection rules. Their signals are always to the blue of the incident beam(s), thus avoiding fluorescence problems. The selection ndes allow one to probe, with optical frequencies, the usual IR spectrum (one photon), not the conventional Raman active vibrations (two photon), but also new vibrations that are synnnetry forbidden in both IR and conventional Raman methods. 

Figure Bl.4.9. Top rotation-tunnelling hyperfine structure in one of the flipping inodes of (020)3 near 3 THz. The small splittings seen in the Q-branch transitions are induced by the bound-free hydrogen atom tiiimelling by the water monomers. Bottom the low-frequency torsional mode structure of the water duner spectrum, includmg a detailed comparison of theoretical calculations of the dynamics with those observed experimentally [ ]. The symbols next to the arrows depict the parallel (A k= 0) versus perpendicular (A = 1) nature of the selection rules in the pseudorotation manifold.

Using the selection rule for allowed transitions the relative intensity for the transition from the state Mg) to Mg+l) is given by... 

The transition between levels coupled by the oscillating magnetic field B corresponds to the absorption of the energy required to reorient the electron magnetic moment in a magnetic field. EPR measurements are a study of the transitions between electronic Zeeman levels with A = 1 (the selection rule for EPR). 

The polarization dependence of the photon absorbance in metal surface systems also brings about the so-called surface selection rule, which states that only vibrational modes with dynamic moments having components perpendicular to the surface plane can be detected by RAIRS [22, 23 and 24]. This rule may in some instances limit the usefidness of the reflection tecluiique for adsorbate identification because of the reduction in the number of modes visible in the IR spectra, but more often becomes an advantage thanks to the simplification of the data. Furthenuore, the relative intensities of different vibrational modes can be used to estimate the orientation of the surface moieties. This has been particularly useful in the study of self-... 

Figure Bl.22.3. RAIRS data in the C-H stretching region from two different self-assembled monolayers, namely, from a monolayer of dioctadecyldisulfide (ODS) on gold (bottom), and from a monolayer of octadecyltrichlorosilane (OTS) on silicon (top). Although the RAIRS surface selection rules for non-metallic substrates are more complex than those which apply to metals, they can still be used to detemiine adsorption geometries. The spectra shown here were, in fact, analysed to yield the tilt (a) and twist (p) angles of the molecular chains in each case with respect to the surface plane (the resulting values are also given in the figure) [40].

Perhaps the best known and most used optical spectroscopy which relies on the use of lasers is Raman spectroscopy. Because Raman spectroscopy is based on the inelastic scattering of photons, the signals are usually weak, and are often masked by fluorescence and/or Rayleigh scattering processes. The interest in usmg Raman for the vibrational characterization of surfaces arises from the fact that the teclmique can be used in situ under non-vacuum enviromnents, and also because it follows selection rules that complement those of IR spectroscopy. 

The conmron flash-lamp photolysis and often also laser-flash photolysis are based on photochemical processes that are initiated by the absorption of a photon, hv. The intensity of laser pulses can reach GW cm or even TW cm, where multiphoton processes become important. Figure B2.5.13 simnnarizes the different mechanisms of multiphoton excitation [75, 76, 112], The direct multiphoton absorption of mechanism (i) requires an odd number of photons to reach an excited atomic or molecular level in the case of strict electric dipole and parity selection rules [117],... 

At this stage, we would like to mention that the model, without the vector potential, is constructed in such a way that it obeys certain selection rules, namely, only the even —> even and the odd —> odd transitions are allowed. Thus any deviation in the results from these selection rules will be interpreted as a symmetry change due to non-adiabatic effects from upper electronic states. 

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